CFD Analysis of Normal Shock using Shock Tube with Five Species

Transcription

1 Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): CFD Analyss of Normal Shock usng Shock Tube wth Fve Speces Shyam S. Kanwar 1, Gaurav Dubey, Mahendra Sngh 3, Gour S. Khanday 4 1,4 Insttute of Technology, Guru Ghasdas Unversty, Department of Mechancal, Kon, Blaspur , Inda Insttute of Technology, Guru Ghasdas Unversty, Department of Industral and Producton Engneerng, Kon, Blaspur , Inda 3 Chrstan College of Engneerng and Technology, Department of Mechancal, Bhla 49006, Inda Abstract: A shock tube s equpment that can produce a movng normal shock wave produced by the sudden burstng of daphragm separatng a hgh pressure gas secton from one at lower pressure gas secton. It s used n hgh speed test facltes. The objectve s to solvng shock tube problem wth CFD code s to calculate the flud flow property and observed how flud parameters are vares wth tme because of shock tube used for hgh speed test and fndng out the behavor of normal shock n the test feld. In the present study, dmensonal Euler equatons are used as governng equaton. The spatal dscretzaton s carred out by unstructured cell-centered Fnte Volume Method. Here the convectve fluxes are evaluated usng Van Leer Flux Splttng Scheme. Speces transport equatons are also added n the Euler equaton for treatment of non-reactng mxng of the gas. Keywords: Shock tube, Movng normal shock, Observaton tme n shock tubes, Euler equatons, Fnte Volume Method, Van Leer Flux Splttng schemes. 1. Introducton Contnuous effort s beng made to ncrease the lmt of maxmum obtanable speed by manknd. Research and development towards ths effort lead us to develop effcent and affordable test facltes. Day by day t s becomng dffcult to gnore the short duraton, hgh speed flow test facltes. Ths thng compelled us and ncreases our curosty to know about the hgh speed flow test facltes. Interest has recently been revved by researchers towards ths hgh speed flow condtons whch can be used to smulate the real condtons encountered by aerospace vehcles n an apparatus whch was frst developed hundred years ago. Today, ths apparatus s known as shock tube. Shock tube s developed to study about the hgh speed flow lke supersonc and Hypersonc related to the aerodynamcs of aerospace vehcles where some concepts such as heat transfer and hgh temperature effects are very mportant. Shock tube s developed to study such phenomena n laboratory condton whch are not possble n actual flght condtons. It s a devce to produce hgh speed flow wth hgh temperatures, by traversng normal shock waves whch are generated by rupture of a daphragm separatng a hgh pressure gas from a low pressure gas. In the 19th century, nterest n the propagaton speeds of flame fronts and detonaton waves led to the constructon of the frst shock tube by Perre Velle n France n Expermental work on shock tube has been carred out n the 1940s n the Unted States and Canada, where ntal experments were made to fnd a method of blast pressure measurement. It was later realzed that shock tubes can be used to nvestgate compressble flow phenomena and extensve experments on nteractons of shocks, rarefactons and contact surfaces were made n Al-Falah [1]. Varous smulaton software and solvers are beng developed n the last decade to facltate varous aerodynamc analyss on supersonc and Hypersonc flow. Extensve analyss s carred out to study and desgn a one dmensonal shock tube wth confguratons of sngle daphragm shock tube. Effect of varaton n pressure rato, composton of drver gas s also studed n ths report. All the analyss s beng done usng L1D (Lnear One Dmensonal) software. The problems are also solved analytcally and solvers are also developed for the study.. Governng Equatons For hgh speed flows, vscous effect s confned to the vcnty of the surface, where the large velocty gradent exsts. Ths regon s known as the boundary layer. Outsde of the boundary layer, the velocty gradents are neglgble resultng n zero shear stress. Ths regon s called the nvscd regon. In the present study, the nvestgaton of soluton procedures for the nvscd flow regon. The governng equaton s known as the Euler equaton []. In two-dmensonal Cartesan coordnates, these can be wrtten as U F G 0 (1) t x y U u v E m u u p F uv uh um u uv uh vm G v p Paper ID:

2 Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): The Euler equatons governng the D flow n the absence of body forces wth speces transport equaton n the conservatve and dfferental form are, ( ) ( u) ( v) 0 t x y ( u) ( u p) ( v) 0 t x y ( E) ( uh) ( vh ) 0 t x y ( m ) ( um ) ( vm ) 0 t x y In the above versons of formulatons, the total specfc energy, E=e+0. 5 (u +v ) the total specfc enthalpy H=h+0. 5 (u +v ), s the mass fracton of the speces gven by m =ρ /ρ. Ths thess consders a soluton to unsteady state Euler equatons and no surface forces are consdered n these equatons. Euler equaton bascally expresses the conservaton of mass, momentum and energy. () (3) (4) (5) of many adjacent control volumes whch cover the doman of nterest. d dt UdV n. Fds 0 (7) V( t) S ( t) The average value of U n a cell wth volume V s 1 U UdV (8) V Eq. 3.1 can be wrtten as d V U n. Fds 0 dt (9) du 1 dt V S() t n. Fds 0 (10) S() t U s the average value of U over the entre control volume, F s the flux vector and n s the unt normal to the surface. And F FI GI j, s the total nvscd flux, upon ntegratng the nvscd flux over the faces of k th control volume the above equaton becomes.1 Equatons of State The governng dfferental equatons for the gas dynamcs (.e. Equaton, Equaton 3 and Equaton 4) are completed by specfyng the thermodynamc propertes of the gas. For a perfect gas, the equaton of state s P RT (6) Where, R s the gas constant. And the speed of sound s gven by a RT The Thermodynamc propertes of a number of deal gases are gven n the Table 1. For the mxtures of gases, the perfect gas relatons can be used together wth the effectve thermodynamc propertes or the mxtures of the gas, can be used the perfect gas relatons together wth the effectve thermodynamc propertes R MR M Where, M s % mass fracton of the speces. Here, the summaton s over the two speces. Table 1: Thermodynamc propertes of deal gases Gas R ( j / kgk ) Ar Helum Ntrogen Argon Co U 1. 0 (11) nf k F nds t Vk 1 k y x Here, n j and s ( x) ( y) s s For the -D ax-symmetrc problems the fnte volume formulaton s gven by du 1 dt V n. Fds 0 (1) S() t 3. Upwnd dscretzaton Upwnd schemes use an adaptve or soluton-senstve fnte dfference stencl to numercally smulate the drecton of propagaton of nformaton n a flow feld. A general form of wrtng any upwnd-type schemes are n1 n at n n n n u u [( u u 1 ) ( u 1 u )] x (13) The upwnd scheme s stable f the followng Courant Fredrch Lewy condton (CFL) condton s satsfed. Here CFL condton s a t / x 1. Scheme s the frst order accurate explct scheme wth only one unknown u n Van Leer flux splttng scheme 3. Numercal Method 3.1 Fnte volume method formulaton The basc dea of a FVM s to satsfy the ntegral form of the conservaton laws to some degree of approxmaton for each The Van Leer scheme tells general flud flow contans wave speeds that are both postve and negatve (so that egenvalue nformaton can pass both upstream and downstream), the basc dea behnd all of these technques s that the flux can be splt nto two components F and F so that each may be properly dscretzed usng relatvely upwnd stencls to mantan stablty and accuracy [3]. Paper ID:

3 Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): If M 1, then F F, F 0 If M 1, then F F, F 0 If 1 M 1 then numercal normal flux F F U F U L R u u u pn x F u v pn y u H um 1 c u M nx c1 M c F v M n y 4 q u c ( 1) M 1 m 3.4 Boundary Condtons Boundary condtons are specfcatons of propertes or condtons on the surfaces of flud domans and sub-domans, and are requred to fully defne the flow smulaton. Boundary condton decdes the soluton of the governng equaton. Fgure 1: Grd for ramp n a channel The lower wall, ncludng the ramp, and the top wall are modeled as beng mpermeable boundares.e. Slp- wall boundary condton s mposed. For computaton, frst order scheme, wth the Van Leer scheme s used for the calculaton of the convectve fluxes. The convergence crteron s based on the dfference n densty values, ρ, at any grd pont between two successve teratons, that s, 10 n1 n 6 Where, n s the teraton ndex. The supersonc flow (Mach=3) at the wall must be tangent to the wall, hence the streamlne at the corner also deflected through the angle15. Due to ths free stream flow s turned nto tself that s the oblque shock whch s clear n present smulaton. From the property of oblque shock theory the flow property must be ncreased. Fgure (a) shows the Mach number dscontnuously decreases. For two dmensonal nvscd flow problem the commonly encountered boundary condtons are, D sold boundary fluxes, Invscd or slp wall boundary condton, Pressure extrapolaton boundary condton, Mrror mage boundary condton, Far feld boundary condton. 4. Solver Valdaton 4.1 Flows through ramp n channel (a) (b) The problem conssts of a unform flow beng dsturbed by a small dsturbance, n ths case the ramp. The supersonc flow through a ramp channel s a standard test case to study the oblque shock for valdaton of D nvscd flow solvers. Free stream condton n ths case gven by Table 1. To start wth Euler computatons, flow at nlet s supersonc, oblque shock wll be generated at the wedge. Usng the geometry the bottom wall s turned upward at the corner through a deflecton angle,15 that s, the corner of concave. The length of the channel s 1. meters and heght s 1 meter. Structured mesh used to have a grd sze of The grd conssts of cells, ponts, faces. The flow geometry and correspondng computatonal mesh are shown n Fgure 1. (c) (d) Fgure : property contours n supersonc flow The densty, temperature and pressure dscontnuously ncreases, whch are demonstrated n Fgure (b), Fgure 3 (c) and Fgure (d) respectvely. In addton, for valdaton, the oblque shock propertes computed from the present solver are n excellent agreement wth the oblque shock relatons [4] as shown n Table 3. Paper ID:

4 Table : Free stream condton for ramp n a channel Test case Free stream nput condtons Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): Γ M U (m/s) P n Pa T n K Table 4: Intal condton for 1D, D and L1D shock tube Table 3: Comparson of varous parameters across shock (wedge) for dfferent speces Parameter Oblque shock theory Solver Result wth =1.66 P /P T /T ρ /ρ M Β Result and Dscusson 5.1 Shock tube problem 1D and D wth varous speces The frst, 1D case, dvde the one dmensonal space of the present smulaton nto 1000 unform grd ponts as shown n Fgure 4 The total length of the smulaton regon s 1 meter, so the ( x=0.01). Fgure 4: Schematc of 1D grd for shock tube (Not to scale) Second, for the D case, the geometry of the shock tube used s as follows: The length of the tube s 1 meter and heght s 0.4 meters. The grd conssts of 6191 cells, 6000 ponts, 1190 faces. The flow geometry and correspondng computatonal mesh are shown n Fgure 5. A shock tube conssts of a long tube, usually of crcular or rectangular cross secton, whch s separated by a thn daphragm nto two parts whch s shown n Fgure 3. One of them, the low pressure chamber, s a feld wth the test gas. The compressed drver gas fed nto the second part, the hgh pressure chamber [5]. The dmenson of the tube can vary. Fgure 5: Schematc of D grd of shock tube Fgure 3: Schematc dagram of shock tube at tme t=0, t=t 1 A shock tube s equpment that can produce a movng normal shock wave produced by the sudden burstng of daphragm separatng a hgh pressure gas secton from one at lower pressure gas secton. It s used n hgh speed test facltes. The objectve s to solve shock tube problem by the solver to determne the flud flow property under the tube. 5. Test condton of shock tube The shock tube problem s the study of the propagaton of shock waves n a one and two dmensonal. Assume a neutral gas exsts n the tube, as there are no charged partcles. At the begnnng, the system s dvded nto two parts wth dfferent pressure and densty. The Intal condton for the soluton s gven by the Table 4. Here two cases are solved by the solver. For computaton, frst order scheme s used for dscrtzaton, whch means that only one step s needed to fnsh a round of computaton. For tme, a unform tme step s adopted for smplcty that s, t= It cannot be sure that ths s rght, after check the result to see whether the tme step s suffcent. If not, change the step. The stablty crteron s based on tme for both the cases, that s, t t CFL t (14) allowed sgnal where, tallowed smallest value for all cells and CFL s the specfed Courant- Fredrch- Lewy number. It s normally restrcted to CFL 1.0. Here the CFL number s set to 0.3. For each cell, the nvcd sgnal tme for 1D and D case s approxmated as Equaton 7 and Equaton 8 respectvely. t sgnal u x j c x y tsgnal u c u c j, j, j, j, j (15) (16) Paper ID:

5 Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): From the Fgure 6, Fgure 7 and Fgure 8 shows the result of shock tube problem (1D and D) at tme t=0.4 ms, together wth the analytcal soluton (L1d, whch gves exact result for shock tube) result of the problem at the same tme [6]. The smulaton result fts the basc confguraton of analytcal result qute well. We can also see clearly that dffusve error s very obvous n some part of the result and all the corners are smoothed. These plots are generated only after the burstng of daphragm. Fgure 9: Daphragm poston at t=0 before burstng Fgure 6: Pressure varaton along a smple shock tube Fgure 10: Pressure contour at t=0.4 ms n shock tube Fgure 11: Densty contour at t=0.4 ms n shock tube after daphragm Fgure 7: Densty varaton along smple shock tube Fgure 8: Temperature varaton along smple shock tube Also assume that there s no moton at the begnnng. Fgure 9 shows the state of the shock tube at t=0from the contour representaton of D shock tube case shows the effects of the movng shock wave propagaton whch s clear n Fgure 10, Fgure 11 and Fgure 1. Fgure 1: Temperature contour at t=0.4 ms n shock tube after daphragm 6. Concluson The work has been carred out keepng n mnd that ths project s about numercal smulaton through dfferent solver and thus desgn of a one dmensonal shock tube wth confguratons of sngle daphragm shock tube s done. The smulaton has been done to reach the supersonc Mach number 3.0 and the solver facltates measurement of the propertes behnd the normal shock wave and ther varaton across the shock. Smulaton s done usng dfferent pressure rato across the shock n order to get wde range of Mach number. From results of CFD smulaton for reflected wave shock tube, one can conclude that Van leer Flux Vector Splttng Scheme cannot be appled for low Mach number Paper ID:

6 Internatonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): flows but can be successfully appled for the supersonc and hypersonc flows. A good match has been found between CFD results and the L1D results. References [1] Al-Falah, A. M.Z. Yusoff, Shaub N.H, T. Yusaf Flow nstablty n shock tube due to shock wave boundary layer contact surface nteractons, a numercal study. European journal of scentfc research, ISSN X vol. 30 No.1 [] Blazek, J. (001). Computatonal Flud Dynamcs Prncpal and Applcatons. Eselver. [3] Lou, M. S. and C. J. Steffen (1999). A New Flux Splttng Scheme. Journal of Computatonal Physcs 107, [4] Anderson, J. D. (1989). Hypersonc and Hgh Temperature Gas Dynamcs. McGraw-Hll Inc. [5] Yadav, S. (01). Numercal Smulaton and Desgnanalyss of Smple Shock Tube For Hypersonc Applcatons. Department of Mecancal. Engneerng, Indan Insttute of Technology Guwahat, Assam (Inda). [6] Jocobs, P. A. (1998). Shock Tube Modelng wth L1d. Department of Mechancal Engneerng, The Unversty of Queensland Reserch Report, 13/98. Author Profle Shyam Sngh kanwar, He has done M.Tech. from Indan Insttute of Technology; Guwahat, Inda, n the year 013. Now he works as an Assstant Professor n the Department of Mechancal Engneerng, IT GGU Unversty (A Central Unversty), Blaspur. Hs research nterest s n the feld of CFD and ts applcaton, Heat Transfer. Gaurav Dubey, Workng as an Assstant Professor n the Department of Industral & producton Engg., Insttute of Technology, GGU Unversty, Blaspur. Hs area of nterest s Sold Mechancs. Gour Shankar Khanday, Workng as an Assstant Professor n the Department of Mechancal Engneerng, Insttute of Technology, GGU Unversty, Blaspur. Hs area of nterest s Machne degn. Paper ID: